Semiconductors — Question 9

Question 9

The minimum conductivity at fixed mass action need not occur at equal carrier densities

At fixed temperature, $np=n_i^2$ and the positive mobilities $\mu_n,\mu_p$ are fixed. Treat n as a variable set by doping and use $\sigma=e(n\mu_n+p\mu_p)$.

1. Find n and p that minimize conductivity.

2. Find the minimum conductivity and compare it with the intrinsic value at n=p=ni.

Question 9 - Solution

The minimum conductivity at fixed mass action need not occur at equal carrier densities

(a) Method and calculation

$$\sigma(n)=e\left(\mu_nn+\mu_pn_i^2/n\right),\quad d\sigma/dn=e(\mu_n-\mu_pn_i^2/n^2).$$$$n_*=n_i\sqrt{\mu_p/\mu_n},\quad p_*=n_i\sqrt{\mu_n/\mu_p},\quad d^2\sigma/dn^2>0.$$

(b) Interpretation and further calculation

$$\sigma_{\min}=2en_i\sqrt{\mu_n\mu_p}.$$$$\sigma_{\mathrm{intrinsic}}-\sigma_{\min}=en_i(\sqrt{\mu_n}-\sqrt{\mu_p})^2\ge0.$$

Physical check

The optimum balances the two conductivity contributions, not necessarily the two carrier densities.